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Theorem qnumdenbi 11715
 Description: Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumdenbi ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶)) ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))

Proof of Theorem qnumdenbi
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 opelxpi 4531 . . . 4 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
213adant1 982 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
3 qredeu 11624 . . . 4 (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))
433ad2ant1 985 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))
5 fveq2 5375 . . . . . . 7 (𝑎 = ⟨𝐵, 𝐶⟩ → (1st𝑎) = (1st ‘⟨𝐵, 𝐶⟩))
6 fveq2 5375 . . . . . . 7 (𝑎 = ⟨𝐵, 𝐶⟩ → (2nd𝑎) = (2nd ‘⟨𝐵, 𝐶⟩))
75, 6oveq12d 5746 . . . . . 6 (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st𝑎) gcd (2nd𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)))
87eqeq1d 2123 . . . . 5 (𝑎 = ⟨𝐵, 𝐶⟩ → (((1st𝑎) gcd (2nd𝑎)) = 1 ↔ ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1))
95, 6oveq12d 5746 . . . . . 6 (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st𝑎) / (2nd𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))
109eqeq2d 2126 . . . . 5 (𝑎 = ⟨𝐵, 𝐶⟩ → (𝐴 = ((1st𝑎) / (2nd𝑎)) ↔ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))))
118, 10anbi12d 462 . . . 4 (𝑎 = ⟨𝐵, 𝐶⟩ → ((((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))) ↔ (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))))
1211riota2 5706 . . 3 ((⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ) ∧ ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩))
132, 4, 12syl2anc 406 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩))
14 op1stg 6002 . . . . . 6 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
15 op2ndg 6003 . . . . . 6 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
1614, 15oveq12d 5746 . . . . 5 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
17163adant1 982 . . . 4 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
1817eqeq1d 2123 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ↔ (𝐵 gcd 𝐶) = 1))
19143adant1 982 . . . . 5 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
20153adant1 982 . . . . 5 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
2119, 20oveq12d 5746 . . . 4 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 / 𝐶))
2221eqeq2d 2126 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) ↔ 𝐴 = (𝐵 / 𝐶)))
2318, 22anbi12d 462 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ ((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶))))
24 riotacl 5698 . . . . . . 7 (∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))) → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ))
25 1st2nd2 6027 . . . . . . 7 ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ) → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨(1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))), (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))))⟩)
263, 24, 253syl 17 . . . . . 6 (𝐴 ∈ ℚ → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨(1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))), (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))))⟩)
27 qnumval 11708 . . . . . . 7 (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
28 qdenval 11709 . . . . . . 7 (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
2927, 28opeq12d 3679 . . . . . 6 (𝐴 ∈ ℚ → ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨(1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))), (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))))⟩)
3026, 29eqtr4d 2150 . . . . 5 (𝐴 ∈ ℚ → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨(numer‘𝐴), (denom‘𝐴)⟩)
3130eqeq1d 2123 . . . 4 (𝐴 ∈ ℚ → ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩ ↔ ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
32313ad2ant1 985 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩ ↔ ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
33 qnumcl 11711 . . . . 5 (𝐴 ∈ ℚ → (numer‘𝐴) ∈ ℤ)
34 qdencl 11712 . . . . 5 (𝐴 ∈ ℚ → (denom‘𝐴) ∈ ℕ)
35 opthg 4120 . . . . 5 (((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ) → (⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))
3633, 34, 35syl2anc 406 . . . 4 (𝐴 ∈ ℚ → (⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))
37363ad2ant1 985 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))
3832, 37bitrd 187 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))
3913, 23, 383bitr3d 217 1 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶)) ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 945   = wceq 1314   ∈ wcel 1463  ∃!wreu 2392  ⟨cop 3496   × cxp 4497  ‘cfv 5081  ℩crio 5683  (class class class)co 5728  1st c1st 5990  2nd c2nd 5991  1c1 7548   / cdiv 8345  ℕcn 8630  ℤcz 8958  ℚcq 9313   gcd cgcd 11483  numercnumer 11704  denomcdenom 11705 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665 This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-sup 6823  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-fz 9684  df-fzo 9813  df-fl 9936  df-mod 9989  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-dvds 11342  df-gcd 11484  df-numer 11706  df-denom 11707 This theorem is referenced by:  qnumdencoprm  11716  qeqnumdivden  11717  divnumden  11719  numdensq  11725
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